Quantum Spin Hall Insulator State in HgTe Quantum Wells

Direct evidence for the dirac-cone topological surface states in the ternary chalcogenide TlBiSe₂

We have performed angle-resolved photoemission spectroscopy on TlBiSe₂, which is a member of the ternary chalcogenides theoretically proposed as candidates for a new class of three-dimensional topological insulators. We found direct evidence for a nontrivial surface metallic state showing an "X"-shaped energy dispersion within the bulk-band gap. The present result unambiguously establishes that TlBiSe₂ is a strong topological insulator with a single Dirac cone at the Brillouin-zone center. The observed bulk-band gap of 0.35 eV is the largest among known topological insulators, making TlBiSe₂ the most promising material for studying room-temperature topological phenomena.

Entanglement spectrum of a disordered topological Chern insulator

We investigate the behavior of a topological Chern insulator in the presence of disorder, with a focus on its entanglement spectrum (EtS) constructed from the ground state. For systems with symmetries, the EtS was shown to contain explicit information about the topological universality class revealed by sorting the EtS against the conserved quantum numbers. In the absence of any symmetry, we demonstrate that statistical methods such as the level statistics of the EtS can be equally insightful, allowing us to distinguish when an insulator is in a topological or trivial phase and to map the boundary between the two phases. The phase diagram of a Chern insulator is explicitly computed as function of Fermi level (EF) and disorder strength using the level statistics of the EtS and energy spectrum, together with a computation of the Chern number (C) via a new, efficient real-space formula.

Half-Heusler compounds as a new class of three-dimensional topological insulators

Using first-principles calculations within density functional theory, we explore the feasibility of converting ternary half-Heusler compounds into a new class of three-dimensional topological insulators (3DTI). We demonstrate that the electronic structure of unstrained LaPtBi as a prototype system exhibits a distinct band-inversion feature. The 3DTI phase is realized by applying a uniaxial strain along the [001] direction, which opens a band gap while preserving the inverted band order. A definitive proof of the strained LaPtBi as a 3DTI is provided by directly calculating the topological Z2 invariants in systems without inversion symmetry. We discuss the implications of the present study to other half-Heusler compounds as 3DTI, which, together with the magnetic and superconducting properties of these materials, may provide a rich platform for novel quantum phenomena.

Hexagonally deformed Fermi surface of the 3D topological insulator Bi2Se3

A hexagonal deformation of the Fermi surface of Bi2Se3 has been for the first time observed by angle-resolved photoemission spectroscopy. This is in contrast to the general belief that Bi2Se3 possesses an ideal Dirac cone. The hexagonal shape is found to disappear near the Dirac node, which would protect the surface state electrons from backscattering. It is also demonstrated that the Fermi energy of naturally electron-doped Bi2Se3 can be tuned by 1% Mg doping in order to realize the quantum topological transport.

Landau quantization of topological surface states in Bi2Se3

We report the direct observation of Landau quantization in Bi2Se3 thin films by using a low-temperature scanning tunneling microscope. In particular, we discovered the zeroth Landau level, which is predicted to give rise to the half-quantized Hall effect for the topological surface states. The existence of the discrete Landau levels (LLs) and the suppression of LLs by surface impurities strongly support the 2D nature of the topological states. These observations may eventually lead to the realization of quantum Hall effect in topological insulators.

Ultrathin topological insulator Bi2Se3 nanoribbons exfoliated by atomic force microscopy

Ultrathin topological insulator nanostructures, in which coupling between top and bottom surface states takes place, are of great intellectual and practical importance. Due to the weak van der Waals interaction between adjacent quintuple layers (QLs), the layered bismuth selenide (Bi(2)Se(3)), a single Dirac-cone topological insulator with a large bulk gap, can be exfoliated down to a few QLs. In this paper, we report the first controlled mechanical exfoliation of Bi(2)Se(3) nanoribbons (>50 QLs) by an atomic force microscope (AFM) tip down to a single QL. Microwave impedance microscopy is employed to map out the local conductivity of such ultrathin nanoribbons, showing drastic difference in sheet resistance between 1-2 QLs and 4-5 QLs. Transport measurement carried out on an exfoliated (<or=5 QLs) Bi(2)Se(3) device shows nonmetallic temperature dependence of resistance, in sharp contrast to the metallic behavior seen in thick (>50 QLs) ribbons. These AFM-exfoliated thin nanoribbons afford interesting candidates for studying the transition from quantum spin Hall surface to edge states.

Spin and charge transport on the surface of a topological insulator

We derive diffusion equations, which describe spin-charge coupled transport on the helical metal surface of a three-dimensional topological insulator. The main feature of these equations is a large magnitude of the spin-charge coupling, which leads to interesting and observable effects. In particular, we predict a new magnetoresistance effect, which manifests in a non-Ohmic correction to a voltage drop between a ferromagnetic spin-polarized electrode and a nonmagnetic electrode, placed on top of the helical metal. This correction is proportional to the cross product of the spin polarization of the ferromagnetic electrode and the charge current between the two electrodes. We also demonstrate tunability of this effect by applying a gate voltage, which makes it possible to operate the proposed device as a transistor.

Interaction-induced criticality in Z(2) topological insulators

We study interaction effects in topological insulators with strong spin-orbit coupling. We find that the interplay of nontrivial topology and Coulomb repulsion induces a novel critical state on the surface of a three-dimensional topological insulator. Remarkably, this interaction-induced criticality, characterized by a universal value of conductivity, emerges without any adjustable parameters. Further, we predict a direct quantum-spin-Hall transition in two dimensions that occurs via a similar critical state.

Confinement-induced berry phase and helicity-dependent photocurrents

The photocurrent in an optically active metal is known to contain a component that switches sign with the helicity of the incident radiation. At low frequencies, this current depends on the orbital Berry phase of the Bloch electrons via the "anomalous velocity" of Karplus and Luttinger. We consider quantum wells in which the parent material, such as GaAs, is not optically active and the relevant Berry phase only arises as a result of quantum confinement. Using an envelope approximation that is supported by numerical tight-binding results, it is shown that the Berry-phase contribution is determined for realistic wells by a cubic Berry phase intrinsic to the bulk material, the well width, and the well direction. These results for the Berry-phase effect suggest that it may already have been observed in quantum well experiments.

Tunable multifunctional topological insulators in ternary Heusler compounds

Recently the quantum spin Hall effect was theoretically predicted and experimentally realized in quantum wells based on the binary semiconductor HgTe (refs 1-3). The quantum spin Hall state and topological insulators are new states of quantum matter interesting for both fundamental condensed-matter physics and material science. Many Heusler compounds with C1(b) structure are ternary semiconductors that are structurally and electronically related to the binary semiconductors. The diversity of Heusler materials opens wide possibilities for tuning the bandgap and setting the desired band inversion by choosing compounds with appropriate hybridization strength (by the lattice parameter) and magnitude of spin-orbit coupling (by the atomic charge). Based on first-principle calculations we demonstrate that around 50 Heusler compounds show band inversion similar to that of HgTe. The topological state in these zero-gap semiconductors can be created by applying strain or by designing an appropriate quantum-well structure, similar to the case of HgTe. Many of these ternary zero-gap semiconductors (LnAuPb, LnPdBi, LnPtSb and LnPtBi) contain the rare-earth element Ln, which can realize additional properties ranging from superconductivity (for example LaPtBi; ref. 12) to magnetism (for example GdPtBi; ref. 13) and heavy fermion behaviour (for example YbPtBi; ref. 14). These properties can open new research directions in realizing the quantized anomalous Hall effect and topological superconductors.

Edge dynamics in a quantum spin Hall state: effects from Rashba spin-orbit interaction

We analyze the dynamics of the helical edge modes of a quantum spin Hall state in the presence of a spatially nonuniform Rashba spin-orbit (SO) interaction. A randomly fluctuating Rashba SO coupling is found to open a scattering channel which causes localization of the edge modes for a weakly screened electron-electron (e-e) interaction. A periodic modulation of the SO coupling, with a wave number commensurate with the Fermi momentum, makes the edge insulating already at intermediate strengths of the e-e interaction. We discuss implications for experiments on edge state transport in a HgTe quantum well.

Second-order nonlinear optical effects of spin currents

Pure spin currents carry information in spintronics and signify novel quantum spin phenomena such as topological insulators. Measuring pure spin currents, however, is difficult since they have no direct electromagnetic induction. Noticing that a longitudinal spin current, in which electrons move along their spin directions, is a chiral quantity, we envisage that it has a chiral sum-frequency optical effect. A systematic symmetry analysis confirms this idea and reveals the second-order optical effects of general spin currents with unique polarization dependence. Microscopic calculations based on the eight-band model of III-V compound semiconductors show that the susceptibility is sizable under realistic conditions. These findings form a basis for "seeing" spin currents where and while they flow with standard nonlinear optical spectroscopy, providing a toolbox to explore a wealth of physics connecting spins and photons.

Gate-controlled one-dimensional channel on the surface of a 3D topological insulator

We investigate the formation of one-dimensional channels on the topological surface under the gate electrode. The energy dispersion of these channels is almost linear in momentum, and its velocity and sign are sensitively dependent on the strength of the gate voltage. Consequently, the local density of states near the gated region has an asymmetric structure with respect to zero energy. In the presence of the electron-electron interaction, the correlation effect can be tuned by the gate voltage. We also suggest a tunneling experiment to verify the presence of these bound states.

Few-layer nanoplates of Bi 2 Se 3 and Bi 2 Te 3 with highly tunable chemical potential

A topological insulator (TI) represents an unconventional quantum phase of matter with insulating bulk band gap and metallic surface states. Recent theoretical calculations and photoemission spectroscopy measurements show that group V-VI materials Bi(2)Se(3), Bi(2)Te(3), and Sb(2)Te(3) are TIs with a single Dirac cone on the surface. These materials have anisotropic, layered structures, in which five atomic layers are covalently bonded to form a quintuple layer, and quintuple layers interact weakly through van der Waals interaction to form the crystal. A few quintuple layers of these materials are predicted to exhibit interesting surface properties. Different from our previous nanoribbon study, here we report the synthesis and characterizations of ultrathin Bi(2)Te(3) and Bi(2)Se(3) nanoplates with thickness down to 3 nm (3 quintuple layers), via catalyst-free vapor-solid (VS) growth mechanism. Optical images reveal thickness-dependent color and contrast for nanoplates grown on oxidized silicon (300 nm SiO(2)/Si). As a new member of TI nanomaterials, ultrathin TI nanoplates have an extremely large surface-to-volume ratio and can be electrically gated more effectively than the bulk form, potentially enhancing surface state effects in transport measurements. Low-temperature transport measurements of a single nanoplate device, with a high-k dielectric top gate, show decrease in carrier concentration by several times and large tuning of chemical potential.

Ballistic quantum spin Hall state and enhanced edge backscattering in strong magnetic fields

The quantum spin Hall (QSH) state, observed in a zero magnetic field in HgTe quantum wells, respects the time-reversal symmetry and is distinct from quantum Hall (QH) states. We show that the QSH state persists in strong quantizing fields and is identified by counterpropagating (helical) edge channels with nonlinear dispersion inside the band gap. If the Fermi level is shifted into the Landau-quantized conduction or valence band, we find a transition between the QSH and QH regimes. Near the transition the longitudinal conductance of the helical channels is strongly suppressed due to the combined effect of the spectrum nonlinearity and enhanced backscattering. It shows a power-law decay B(-2N) with magnetic field B, determined by the number of backscatterers on the edge N. This suggests a rather simple and practical way to probe the quality of quasiballistic QSH devices using magnetoresistance measurements.

Collective modes of a helical liquid

We study low energy collective modes and transport properties of the "helical metal" on the surface of a topological insulator. At low energies, electrical transport and spin dynamics at the surface are exactly related by an operator identity equating the electric current to the in-plane components of the spin degrees of freedom. From this relation it follows that an undamped spin wave always accompanies the sound mode in the helical metal-thus it is possible to "hear" the sound of spins. In the presence of long range Coulomb interactions, the surface plasmon mode is also coupled to the spin wave, giving rise to a hybridized "spin-plasmon" mode. We make quantitative predictions for the spin-plasmon in Bi2Se3, and discuss its detection in a spin-grating experiment.

Magnetic doping and kondo effect in bi(2)se(3) nanoribbons

A simple surface band structure and a large bulk band gap have allowed Bi2Se3 to become a reference material for the newly discovered three-dimensional topological insulators, which exhibit topologically protected conducting surface states that reside inside the bulk band gap. Studying topological insulators such as Bi2Se3 in nanostructures is advantageous because of the high surface-to-volume ratio, which enhances effects from the surface states; recently reported Aharonov-Bohm oscillation in topological insulator nanoribbons by some of us is a good example. Theoretically, introducing magnetic impurities in topological insulators is predicted to open a small gap in the surface states by breaking time-reversal symmetry. Here, we present synthesis of magnetically doped Bi2Se3 nanoribbons by vapor-liquid-solid growth using magnetic metal thin films as catalysts. Although the doping concentration is less than approximately 2%, low-temperature transport measurements of the Fe-doped Bi2Se3 nanoribbon devices show a clear Kondo effect at temperatures below 30 K, confirming the presence of magnetic impurities in the Bi2Se3 nanoribbons. The capability to dope topological insulator nanostructures magnetically opens up exciting opportunities for spintronics.

Aharonov-Bohm interference in topological insulator nanoribbons

Topological insulators represent unusual phases of quantum matter with an insulating bulk gap and gapless edges or surface states. The two-dimensional topological insulator phase was predicted in HgTe quantum wells and confirmed by transport measurements. Recently, Bi(2)Se(3) and related materials have been proposed as three-dimensional topological insulators with a single Dirac cone on the surface, protected by time-reversal symmetry. The topological surface states have been observed by angle-resolved photoemission spectroscopy experiments. However, few transport measurements in this context have been reported, presumably owing to the predominance of bulk carriers from crystal defects or thermal excitations. Here we show unambiguous transport evidence of topological surface states through periodic quantum interference effects in layered single-crystalline Bi(2)Se(3) nanoribbons, which have larger surface-to-volume ratios than bulk materials and can therefore manifest surface effects. Pronounced Aharonov-Bohm oscillations in the magnetoresistance clearly demonstrate the coherent propagation of two-dimensional electrons around the perimeter of the nanoribbon surface, as expected from the topological nature of the surface states. The dominance of the primary h/e oscillation, where h is Planck's constant and e is the electron charge, and its temperature dependence demonstrate the robustness of these states. Our results suggest that topological insulator nanoribbons afford promising materials for future spintronic devices at room temperature.

CT-invariant quantum spin Hall effect in ferromagnetic graphene

We predict a quantum spin Hall effect (QSHE) in ferromagnetic graphene under a magnetic field. Unlike the previous QSHE, this QSHE appears in the absence of spin-orbit interaction and thus, is arrived at from a different physical origin. The previous QSHE is protected by the time-reversal (T) invariance. This new QSHE is protected by CT invariance, where C is the charge conjugation operation. Because of this QSHE, the longitudinal resistance exhibits quantum plateaus. The plateau values are at 1/2, 1/6, 3/28, ..., (in units of h/e2), depending on the filling factors of the spin-up and spin-down carriers. The spin Hall resistance is also investigated and is found to be robust against the disorder.

Unconventional superconductivity on a topological insulator

We study proximity-induced superconductivity on the surface of a topological insulator (TI), focusing on unconventional pairing. We find that the excitation spectrum becomes gapless for any spin-triplet pairing, such that both subgap bound states and Andreev reflection is strongly suppressed. For spin-singlet pairing, the zero-energy surface state in the d(xy)-wave case becomes a Majorana fermion, in contrast with the situation realized in the topologically trivial high-T(c) cuprates. We also study the influence of a Zeeman field on the surface states. Both the magnitude and direction of this field are shown to strongly influence the transport properties, in contrast with the case without TI. We predict an experimental signature of the Majorana states via conductance spectroscopy.

Electron teleportation via Majorana bound states in a mesoscopic superconductor

Zero-energy Majorana bound states in superconductors have been proposed to be potential building blocks of a topological quantum computer, because quantum information can be encoded nonlocally in the fermion occupation of a pair of spatially separated Majorana bound states. However, despite intensive efforts, nonlocal signatures of Majorana bound states have not been found in charge transport. In this work, we predict a striking nonlocal phase-coherent electron transfer process by virtue of tunneling in and out of a pair of Majorana bound states. This teleportation phenomenon only exists in a mesoscopic superconductor because of an all-important but previously overlooked charging energy. We propose an experimental setup to detect this phenomenon in a superconductor-quantum-spin-Hall-insulator-magnetic-insulator hybrid system.

Superconductivity in CuxBi2Se3 and its implications for pairing in the undoped topological insulator

Bi2Se3 is one of a handful of known topological insulators. Here we show that copper intercalation in the van der Waals gaps between the Bi2Se3 layers, yielding an electron concentration of approximately 2x10{20} cm{-3}, results in superconductivity at 3.8 K in CuxBi2Se3 for 0.12<or=x<or=0.15. This demonstrates that Cooper pairing is possible in Bi2Se3 at accessible temperatures, with implications for studying the physics of topological insulators and potential devices.

Tuning the conductance of dirac fermions on the surface of a topological insulator

We study the transport properties of the Dirac fermions with a Fermi velocity v{F} on the surface of a topological insulator across a ferromagnetic strip providing an exchange field J over a region of width d. We show that the conductance of such a junction, in the clean limit and at low temperature, changes from oscillatory to a monotonically decreasing function of d beyond a critical J. This leads to the possible realization of a magnetic switch using these junctions. We also study the conductance of these Dirac fermions across a potential barrier of width d and potential V0 in the presence of such a ferromagnetic strip and show that beyond a critical J, the criteria of conductance maxima changes from chi=eV{0}d/variant Planck's over v{F}=npi to chi=(n+1/2)pi for integer n. We point out that these novel phenomena have no analogs in graphene and suggest experiments which can probe them.

Experimental demonstration of topological surface states protected by time-reversal symmetry

We report direct imaging of standing waves of the nontrivial surface states of topological insulator Bi2Te3 using a scanning tunneling microscope. The interference fringes are caused by the scattering of the topological states off Ag impurities and step edges on the Bi2Te3(111) surface. By studying the voltage-dependent standing wave patterns, we determine the energy dispersion E(k), which confirms the Dirac cone structure of the topological states. We further show that, very different from the conventional surface states, backscattering of the topological states by nonmagnetic impurities is completely suppressed. The absence of backscattering is a spectacular manifestation of the time-reversal symmetry, which offers a direct proof of the topological nature of the surface states.

Topological phonon modes and their role in dynamic instability of microtubules

Microtubules (MTs) are self-assembled hollow protein tubes playing important functions in live cells. Their building block is a protein called tubulin, which self-assembles in a particulate 2 dimensional lattice. We study the vibrational modes of this lattice and find Dirac points in the phonon spectrum. We discuss a splitting of the Dirac points that leads to phonon bands with nonzero Chern numbers, signaling the existence of topological vibrational modes localized at MTs edges, which we indeed observe after explicit calculations. Since these modes are robust against the large changes occurring at the edges during the dynamic cycle of the MTs, we can build a simple mechanical model to illustrate how they would participate in this phenomenon.

Detecting the Majorana fermion surface state of 3He-B through spin relaxation

The Majorana fermion, which can be useful for topological quantum computation, has eluded detection. (3)He-B, recently shown to be a time-reversal invariant topological superfluid, has a gapless Majorana fermion surface state. We show here that an electron spin relaxation experiment can detect this surface state--its Majorana nature through the Zeeman field direction dependence of the relaxation time 1/T(1) proportional, proportional to sin(2)theta, where theta is the angle between the field and the surface normal. We propose an experimental setup where an electron inside a nanobubble is injected below the (3)He liquid surface.

Three-dimensional topological insulators on the pyrochlore lattice

Electrons hopping on the sites of a three-dimensional pyrochlore lattice are shown to form topologically nontrivial insulating phases when the spin-orbit (SO) coupling and lattice distortions are present. Of 16 possible topological classes 9 are realized for various parameters in this model. Specifically, at half-filling an undistorted pyrochlore lattice with a SO term yields a "pristine" strong topological insulator with a Z(2) index (1;000). At quarter filling various strong and weak topological phases are obtained provided that both SO coupling and uniaxial lattice distortion are present. Our analysis suggests that many of the nonmagnetic insulating pyrochlores could be topological insulators.

Theory of the topological anderson insulator

We present an effective medium theory that explains the disorder-induced transition into a phase of quantized conductance, discovered in computer simulations of HgTe quantum wells. It is the combination of a random potential and quadratic corrections proportional to p2 sigma(z) to the Dirac Hamiltonian that can drive an ordinary band insulator into a topological insulator (having an inverted band gap). We calculate the location of the phase boundary at weak disorder and show that it corresponds to the crossing of a band edge rather than a mobility edge. Our mechanism for the formation of a topological Anderson insulator is generic, and would apply as well to three-dimensional semiconductors with strong spin-orbit coupling.

Fractional topological insulators

We analyze generalizations of two-dimensional topological insulators which can be realized in interacting, time reversal invariant electron systems. These states, which we call fractional topological insulators, contain excitations with fractional charge and statistics in addition to protected edge modes. In the case of s(z) conserving toy models, we show that a system is a fractional topological insulator if and only if sigma(sH)/e* is odd, where sigma(sH) is the spin-Hall conductance in units of e/2pi, and e* is the elementary charge in units of e.

Two interacting helical edge modes in quantum spin Hall systems

We study theoretically two interacting one-dimensional helical modes at the edges of the quantum spin Hall systems. A new type of interedge correlated liquid without a spin gap is found when the intraedge repulsion is stronger than the interedge repulsion. This liquid shows a diverging density wave and superconductivity correlations much stronger than those of the spinfull electrons. Possible experimental observations are also discussed.

Manifestation of helical edge states as zero bias magneto-tunneling-conductance peaks in noncentrosymmetric superconductors

Helical edge states exist in the mixed spin singlet and triplet phase of a noncentrosymmetric superconductor (NCSS) when the pair amplitude (PA) in the negative helicity band, Δ(-), is smaller than the PA in the positive helicity band, Δ(+), i.e., when the PA in the triplet component is more than the same in the singlet component. We numerically determine energies of these edge states as a function of γ = Δ(-)/Δ(+). The presence of these edge states is reflected in the tunneling process from a normal metal to an NCSS across a bias energy eV. (i) Angle resolved spin conductance (SC) obeying the symmetry g(s)(φ) = -g(s)(-φ) shows peaks when the bias energy equals the available quasiparticle edge state energy provided [Formula: see text]. (ii) The total SC, G(s), is zero but modulates with eV for finite magnetic field H. (iii) The zero bias peaks of G(s) and total charge conductance, G(c), at finite H split into two at finite eV for moderate H. (iv) At zero bias, G(c) and G(s) increase with H and show peaks at |H|∼γH(0), where H(0) is a characteristic field.

Manipulation of the Majorana fermion, Andreev reflection, and Josephson current on topological insulators

We study theoretically the transport properties of a normal metal (N)/ferromagnet insulator (FI)/superconductor (S) junction and a S/FI/S junction formed on the surface of a three-dimensional topological insulator, where the chiral Majorana mode exists at the FI/S interface. We find the chiral Majorana mode generated in N/FI/S and S/FI/S junctions is very sensitively controlled by the direction of the magnetization m in the FI region. In particular, the current-phase relation of the Josephson current in S/FI/S junctions has a phase shift of neither 0 nor pi that can be tuned continuously by the component of m perpendicular to the interface.

Topological insulators and nematic phases from spontaneous symmetry breaking in 2D fermi systems with a quadratic band crossing

We investigate the stability of a quadratic band-crossing point (QBCP) in 2D fermionic systems. At the noninteracting level, we show that a QBCP exists and is topologically stable for a Berry flux +/-2pi if the point symmetry group has either fourfold or sixfold rotational symmetries. This putative topologically stable free-fermion QBCP is marginally unstable to arbitrarily weak short-range repulsive interactions. We consider both spinless and spin-1/2 fermions. Four possible ordered states result: a quantum anomalous Hall phase, a quantum spin Hall phase, a nematic phase, and a nematic-spin-nematic phase.

A tunable topological insulator in the spin helical Dirac transport regime

Helical Dirac fermions-charge carriers that behave as massless relativistic particles with an intrinsic angular momentum (spin) locked to its translational momentum-are proposed to be the key to realizing fundamentally new phenomena in condensed matter physics. Prominent examples include the anomalous quantization of magneto-electric coupling, half-fermion states that are their own antiparticle, and charge fractionalization in a Bose-Einstein condensate, all of which are not possible with conventional Dirac fermions of the graphene variety. Helical Dirac fermions have so far remained elusive owing to the lack of necessary spin-sensitive measurements and because such fermions are forbidden to exist in conventional materials harbouring relativistic electrons, such as graphene or bismuth. It has recently been proposed that helical Dirac fermions may exist at the edges of certain types of topologically ordered insulators-materials with a bulk insulating gap of spin-orbit origin and surface states protected against scattering by time-reversal symmetry-and that their peculiar properties may be accessed provided the insulator is tuned into the so-called topological transport regime. However, helical Dirac fermions have not been observed in existing topological insulators. Here we report the realization and characterization of a tunable topological insulator in a bismuth-based class of material by combining spin-imaging and momentum-resolved spectroscopies, bulk charge compensation, Hall transport measurements and surface quantum control. Our results reveal a spin-momentum locked Dirac cone carrying a non-trivial Berry's phase that is nearly 100 per cent spin-polarized, which exhibits a tunable topological fermion density in the vicinity of the Kramers point and can be driven to the long-sought topological spin transport regime. The observed topological nodal state is shown to be protected even up to 300 K. Our demonstration of room-temperature topological order and non-trivial spin-texture in stoichiometric Bi(2)Se(3).M(x) (M(x) indicates surface doping or gating control) paves the way for future graphene-like studies of topological insulators, and applications of the observed spin-polarized edge channels in spintronic and computing technologies possibly at room temperature.

Topological insulator: a new quantized spin Hall resistance robust to dephasing

The dephasing effect on the quantum spin Hall effect (QSHE) is studied. Without dephasing, the longitudinal resistance in a QSHE system exhibits the quantum plateaus. We find that these quantum plateaus are robust against the normal dephasing but fragile with the spin dephasing. Thus, these quantum plateaus survive only in mesoscopic samples. Moreover, the longitudinal resistance increases linearly with the sample length but is insensitive to the sample width. These characters are in excellent agreement with the recent experimental results [Science 318, 766 (2007)10.1126/science.1148047]. In addition, we define a new spin Hall resistance that also exhibits quantum plateaus. In particular, these plateaus are robust against any type of dephasing and therefore, survive in macroscopic samples and better reflect the topological nature of QSHE.

Tight-binding modeling and low-energy behavior of the semi-Dirac point

We develop a tight-binding model description of semi-Dirac electronic spectra, with highly anisotropic dispersion around point Fermi surfaces, recently discovered in electronic structure calculations of VO2-TiO2 nanoheterostructures. We contrast their spectral properties with the well-known Dirac points on the honeycomb lattice relevant to graphene layers and the spectra of bands touching each other in zero-gap semiconductors. We also consider the lowest order dispersion around one of the semi-Dirac points and calculate the resulting electronic energy levels in an external magnetic field. In spite of apparently similar electronic structures, Dirac and semi-Dirac systems support diverse low-energy physics.